* Corresponding author: [email protected] Inventory control systems for stochastic lead time demand Mustafid 1,2,* , Dwi Ispriyanti 3 , Sugito 4 , and Diah Safitri 5 1 Master Program of Information Systems, School of Postgraduate Studies, Diponegoro University, Semarang - Indonesia 2,3,4,5 Department of Statistics, Faculty of Sains and Mathematic, Diponegoro University, Semarang – Indonesia Abstract. Product inventory control can be used to provide information related to processes of production and distribution products based on lead time and product demand from consumers. The research aims to design product inventory control systems for stochastic lead time demand, and using exponential smoothing to make lead time demand prediction. The design of the control system uses the probability method based on the estimated number of product demand from the market or consumer. This research provides theoretical framework for control systems of products that can determine the size of inventory parameters in supply chain management, that is reorder point and fixed order quantity, so it can manage the production amount and supply of products according to market or consumer demand. The benefits of a product control systems in the inventory management are to provide the necessary inventory allocation information in making decisions to determine the amount of production and distribution of products to consumers. Keywords: Probabilistic method; inventory control systems; stochastic lead time demand 1 Introduction In highly competitive markets, all manufacturers seek to improve the quality of their industrial products, by reducing product and service costs, and to shorten product delivery and distribution time to consumers [1]. In the era of quality and business competition is very tight, the life cycle of clothing products to the consumer demands with a shorter time and has an element of uncertainty. This can lead to a high risk in the clothing supply chain management industry [2]. Currently, the clothing industry market changes increasingly dynamic and complex, and must face market changes every day. This should be a management action to be able to predict and prepare the business for each day that changes [3]. Dynamic changes in the clothing industry must be followed by inventory management that can anticipate business processes that require various stages, and each stage is accompanied by an element of uncertainty caused by uncertain market demand. In order for the clothing industry to succeed in a highly competitive business environment, clothing industry must be able to run supply chain inventory management in order to improve supply chain performance continuously [4]. The inventory control model (ICM) is part of the supply chain inventory management that plays an important role in controlling the company resources in all links by paying attention to the quality of each supplier link to the customer [5, 6]. Many companies are faced with a rigorous, competitive environment and emerging uncertainties as a result of information technology innovation and changing customer needs. Inventory control (IC) can be used for consistent and sustainable performance measurement of each link in the supply chain that makes a key role in the success of a company to achieve a business target goal with a certain profitability [7]. The ICM must be adaptive to adapt to changes in market or consumer demand. As an example of a higher level of consumer demand from inventory will have an impact on the aspect of customer satisfaction to get the product according to the desired time. In this case, ICM aims to overcome the existence of deficiency or excess product in accordance with production capacity and inventory [8]. Probabilistic and statistical approaches for inventory control systems (ICS) in the clothing industry focus on two variables, that is variables of demand and lead time. Since both variables are stochastic, modeling the IC using the probability distribution assumption of both variables. The use of assumptions for daily demand has a normal distribution can yield more suitable results, but the assumption of a normal distribution of demand data is often not appropriate, especially if demand per unit of time or lead time are fluctuating or has a large variance. The compound distribution of demand during the lead time is use by the distribution of demand and the distribution of lead times. The probability distribution of the combined variables can be normal or other https://doi.org/10.1051/e3sconf/201873 , (2018) E3S Web of Conferences 73 ICENIS 2018 0 1 1 30 3 2 21 1 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
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Inventory control systems for stochastic lead time
demandMustafid1,2,*, Dwi Ispriyanti3, Sugito4, and Diah
Safitri5
1Master Program of Information Systems, School of Postgraduate Studies, Diponegoro University, Semarang - Indonesia 2,3,4,5Department of Statistics, Faculty of Sains and Mathematic, Diponegoro University, Semarang – Indonesia
Abstract. Product inventory control can be used to provide information related to processes of production
and distribution products based on lead time and product demand from consumers. The research aims to
design product inventory control systems for stochastic lead time demand, and using exponential
smoothing to make lead time demand prediction. The design of the control system uses the probability
method based on the estimated number of product demand from the market or consumer. This research
provides theoretical framework for control systems of products that can determine the size of inventory
parameters in supply chain management, that is reorder point and fixed order quantity, so it can manage the
production amount and supply of products according to market or consumer demand. The benefits of a
product control systems in the inventory management are to provide the necessary inventory allocation
information in making decisions to determine the amount of production and distribution of products to
consumers.
1 Introduction
In the era of quality and business competition is very
tight, the life cycle of clothing products to the consumer
demands with a shorter time and has an element of
uncertainty. This can lead to a high risk in the clothing
supply chain management industry [2].
Currently, the clothing industry market changes
increasingly dynamic and complex, and must face
market changes every day. This should be a management
action to be able to predict and prepare the business for
each day that changes [3]. Dynamic changes in the
clothing industry must be followed by inventory
management that can anticipate business processes that
require various stages, and each stage is accompanied by
an element of uncertainty caused by uncertain market
demand.
In order for the clothing industry to succeed in a
highly competitive business environment, clothing
industry must be able to run supply chain inventory
management in order to improve supply chain
performance continuously [4]. The inventory control
model (ICM) is part of the supply chain inventory
management that plays an important role in controlling
the company resources in all links by paying attention to
the quality of each supplier link to the customer [5, 6].
Many companies are faced with a rigorous, competitive
environment and emerging uncertainties as a result of
information technology innovation and changing
customer needs. Inventory control (IC) can be used for
consistent and sustainable performance measurement of
each link in the supply chain that makes a key role in the
success of a company to achieve a business target goal
with a certain profitability [7].
The ICM must be adaptive to adapt to changes in
market or consumer demand. As an example of a higher
level of consumer demand from inventory will have an
impact on the aspect of customer satisfaction to get the
product according to the desired time. In this case, ICM
aims to overcome the existence of deficiency or excess
product in accordance with production capacity and
inventory [8].
control systems (ICS) in the clothing industry focus on
two variables, that is variables of demand and lead time.
Since both variables are stochastic, modeling the IC
using the probability distribution assumption of both
variables.
normal distribution can yield more suitable results, but
the assumption of a normal distribution of demand data
is often not appropriate, especially if demand per unit of
time or lead time are fluctuating or has a large variance.
The compound distribution of demand during the lead
time is use by the distribution of demand and the
distribution of lead times. The probability distribution of
the combined variables can be normal or other
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
distributions according to the data characteristics of daily
demand and lead time periodically.
The research aims to apply ICS on the predicted lead
time demand (LTD) using probabilistic and statistics
approach. Prediction method in the research using
exponential smoothing approach. The ICS can be used to
determine the numbers of safety stock (SS) and reorder
point (ROP) for planning of product inventory renewal
according to consumer demand.
study of IC with deterministic demand data has been
done by [6] and [9]. The supply chain system framework
is based on information flow in the form of product
demand, and material flow of products has been
developed by [7]. The concept of supply chain system
can be adjusted to be the basis in modeling inventory
control product.
The research designs ICS that can be used to analyze
SS and ROP based on the combined of demand and lead
time stochasticlly. The material studied in this research
includes the use of probabilistic and statistical
approaches in ICM. In this case study, the data used
were not normally distributed, so the exponential
smoothing approach use for transformation into normal
distributed data, and also as a prediction model in the
ICM based on previous actual data of demand.
The variables in this research use the number of daily
demand and lead times, taking a case study in the
clothing industry. Data of daily demand in the form of
historical data from daily demand. The observational
data a sample were taken for 105 days with 8 lead times
from April to June 2016. Daily demand data and lead
times are define as stochastic variables.
The variability of demand and lead time are the basic
for making theoretical development of theory and its
application in IC. SS determination uses a combined of
daily demand and lead time into variables of LTD. In
statistical theory, combined variables are called
compound variables. The LTD is a random variable, in
which those distribution is determined by the combined
of the distribution of demand variable and lead variable.
Data of demand from consumers or transactions with
consumers in retail are monitored through demand data
on supply chains sent by retail every day. The demand
data is input to the system, then processed to produce the
output of estimation of SS and ROP as the basis for
performing the amount of stock order that is useful for
arranging product quantities and product distribution
according to the number of demand.
3 Stochastic lead time demand
The ICM is designed using variable of LTD based on
demand on the lead time from consumers or retail. Let D
be a srandom variable in the form of a daily demand in a
periodic lead time L. Daily demand (D) and lead time (L)
are stochastic random variables. Daily demand and lead
time are assumed to be random variables that are
mutually independent and distributed identically. The
random variable of demand D is also assumed to be
independent with the lead time L. LTD X at time L is
expressed by:
X = (1)
Suppose mean and variance of D and L are respectively
expressed by:
The random variables of D and L are can be
estimated using observation data. The mean values of (2)
and (3) can be estimated by data observations:
= (5)
= (6)
Similarly, the variance values of (2) and (3) can be
estimated by:
= (7)
= (8)
whrere D1, D 2 , …, D N are data of demand with N = L1
+ L2 + · · · + Lk, Li is lead time in i period, i = 1, …, k.
Using the unbiased properties of variable of demand
D and lead time L, then by (5), (6), (7) and (8) obtained:
E( = dan E( ) =
E( = dan E( ) =
the mean and variance values of observational data
(samples) with formulas (5), (6), (7) and (8).
The variable X in equation (1) can be define as a
compound random variable, and mean and variance of X
can be derived by:
= (10)
In certain cases, if D and L have a normal distribution,
then distribution of X can be determined as a normal
distribution. For example, Di is random variable having a
normal distribution with mean and variance , , it
can be proved that X as a new random variable following
the normal distribution with mean and variance
.
The prediction of the number of demand in a certain
period is based on the forecast data from the previous
demand. The prediction model for next demand use
exponential smoothing approach of the demand during
period T [10] with illustration as in Fig. 1 and Fig 2.
Let is the daily demand as stochastic process at
time t. Predicted daily demand D for the next one (Fig 1)
is calculated based on the previous time period T using
exponential smoothing with the formula:
/ T (11)
where is the demand prediction for time t + 1, is
the actual demand at time t, and α is the parameter which
is the weight of the previous demand. The weight value
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
2
of α is derived for data is normal distribution with a low
error.
Fig. 2. Demand prediction for next lead time
Furthermore, using (11) can be calculate the
prediction of demand for period t to lead time L, (or tj +
Lj) by formula:
Based on (12), we can define prediction of all daily
demand with k lead time by:
= (13)
The value of mean and variance from calculated in the
same way by the formula (9) and (10).
The computational results for parameters of the
demand and lead time using the formula (5), (6), (7), (8)
and (9) are given in Table 1. The profile of actual daily
demand data and the smoothing data predicted with
exponential smoothing are given in Fig, 3. Based on the
profile in Fig 3, show that variable of demand and lead
time are probabilistic. Using the formula (12) and (13),
we can calculate the value of total and average of LTD
prediction based on all daily demand and all lead time as
a basis for determining ROP and ROP,
Table 1. Parameter of lead time and LTD
Parameter Lead
satisfies the assumption of mutually independent and
distributed identical. The probability approach is used to
find the probability distribution of LTD. To prove that
the data of demand and lead time have a normal
distribution is used the Kolmogorov-Smirnov test. The
result of normal distribution hypothesis test for actual
demand data gives p-value = 0,026, which states that
actual data of demand is not normally distributed for α =
0,05. Therefore, it is necessary to refine the data with
exponential smoothing method using formula (11).
The smoothing of demand data with exponential
smoothing (11) is given in Fig.3, and based on
Kolmogorov-Smirnov test for the smoothing of demand
gives p-value = 0,200. Using α = 0.05, and since p-value
= 0.200> α = 0.05, thus accepting the hypothesis that the
data of smoothing demand/ predicted of demand is
normally distributed. Likewise for the observation data
lead time, obtained p-value = 0.200, so that lead time is
also normally distributed.
The prediction model with exponential smoothing (11)
can be used for the smoothing model as a transformation
model for normal distribution and as a prediction model.
The prediction accuracy can be determined by the
MAPE (Mean Absolute Percentage Error) method, with
the formula:
MAPE =
where N is the number of daily demand data used for the
prediction model. The computational results of the
predicted data in chart form are given in Fig. 3.
Furthermore, the computational results of MAPE show
that for the prediction of demand obtained MAPE = 45%
(reasonable), with for predicted result for LTD obtained
MAPE = 6% (high accuracy). Although MAPE for data
demand data prediction is quite large, but for data
prediction of LTD is very small.
5 Inventory control systems
information system (SCIS) framework with daily
demand and lead times as the input of systems. Daily
demand and lead time is used to make predictions of
product needs as materials to plan the number of
products to be produced. The prediction model use
exponential smoothing method with equation (11).
The design of ICS for the clothing industry are given
in Fig. 4, with system inputs in the form of daily demand
and lead times (Fig 3). Procedures in access to data input
for the system, the input system in the form of demand
from consumers, initially accommodated in retail, so
accessed into the database system conducted by
administrators and retail. The system database is
designed as an integrated database within ICS.
Fig. 4 shows that the consumer or retail demand data
is the basis for making predictions of the number of
products produced. Predicted product quantities and
product inventory estimates are determined on the basis
of SS and ROP as parameters in IC. Web administrators
have a master data entry facility that is used to estimate
the management of product stock for raw material
planning required in producing a number of planned
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
as SMS / WA every day.
Fig. 4. Inventory control systems
The ICS is defined as a tool for the management and
control of supply, storage, distribution, and recording of
products to maintain adequate stock quantities in
accordance with consumer demand or needs. With this
control system is expected to generate maximum profits
with minimal inventory investment without reducing the
level of customer satisfaction or the level of demand or
order filling. Three parameters used for control
inventory:
Probability approach with normal distribution is used to
make ICM with input data of demand and lead time in
the form of random and stochastic variable. To state that
the stochastic process or random sample data satisfy the
assumption of normal distribution will be use by
Kolmogorov-Smirnov test. With normal distribution
function it can be used to determine SS and ROP as
given in Fig. 5.
Using the properties of the normal probability for
random variable X, then expecting X ≤ ROP, it can be
determined the probability that LTD is less than ROP
with the least probability of 1 - α:
P(X ≤ ROP) ≥ 1 - α
as given in Fig. 5. If using α = 0.05 or 5%, then
probability for LTD is less than ROP given in the
formula:
Therefore, the level of confidence that inventory for
demand is always over 95%. Using the properties of the
confidence interval on the normal distribution curve
(Fig. 5), it is obtained that
ROP = E(X) +
and using formula (16) with α = 0,05, = 1,65,
obtained:
SS =
= 431.
ROP = +
= 1632.
The result of computing for parameter of the ICS is
visualized with the output of a normal distribution curve
showing the mean, SS and ROP as given in Fig. 6.
Fig. 6. Output for SS and ROP
In IC on the clothing supply chain industry, stock
supply monitoring on the supply chain is based on
checking on the current stock status. Current stock is
calculated by:
Current stock = initial stock - total demand
Based on demand data, then at the end of the lead time
can be done the latest stock calculation (end), that is the
initial stock at the beginning of lead time minus the
number of LTD. Furthermore, the latest stock results
compared to SS.
a. No inventory is procured, if the current inventory
number is greater than the number of SS of the
product : IP – LTD > SS.
b. If the current inventory number is smaller than the
number of SS inventory, a procurement demand will
be made, with quantity demand Q is determined by
formula:
where IP is the current stock position, LTD is the
number of LTD, SS is the SS of the product, whereas Q
is the number of product demand. Using the formula
(17), the number of stocks delivery is as follows:
Q = SS -IP + LTD
Stock monitoring is done by looking at the latest
stock system output at lead times compared to SS. If the
current stock is larger than the SS, then the stock is in a
safety condition, meaning no stock renewal is made. On
the contrary, if current stock is smaller than SS, then
stock is in watch condition, that means stock updating
must be done soon, with minimum stock is ROP. In this
case research, for the last lead time it appears that the
current stock is 266 pcs. By comparing the SS = 431, the
current stock is smaller than SS. So as soon as the stock
renewal is equal to Q = 1483 for the next lead time.
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
demand data as a condition of using SS formula (16),
while stocks are safety determinant factor for calculating
stock renewal. The use of the probability method for lead
time demend modeling with a more accurate estimation
of inventory parameters should take into account all
available demand data within a given time period of a
stochastically. In the traditional approach, demand
parameters are estimated taking into consideration only
with demand during the lead period. The length of lead
time can be either constant or deterministic, but the
beginning of the refill process is not known before, so
simply using the demand during the lead time alone can
cause a very bad estimate.
Because demand is stochastic, the number of demand
often changes. This has an impact on SS, ie that SS tend
to be low when demand intensity is low, and sometimes
high if demand intensity is high. To overcome this, the
determination of SS is expected to overcome the
deviation (demand) on a particular day that often occurs.
Many IC research by establishing a safe stock use the
assumption that demand and lead times are deterministic
[10, 11]. Similarly, the number of demand for any given
time period is not known in advance, either the number
of demand or the time of the demand. The problem has
an impact on the results of the analysis that is not
necessarily realistic. Other obstacles in IC, product
delivery time can be uneven and uncertain, for several
reasons, such as the amount of demand with erratic time,
quality and transportation problems.
theory on ICS designed within the framework of SCIS
with probability approach. This result is consistent with
the results of previous research. For example [12] which
states that studying the behavior of demand and lead
time is very important to achieve a useful system
representation in order to make the right decision.
6 Conclusion
framework using input demand and lead time. As a case
study, Inventory control systems can be used to
determine the number of safety stock and reorder point
of lead time demand according to market or consumer
demand. Determination of safety stock and reorder point
using normal distribution curve.
amount to perform a stock renewal.
References
of Operations Management 33, 1 (2015).
2. A. Agarwal, R. Shankar, M.K. Tiwari. Industrial
Marketing Management 36 (2007).
Computer Security 16, 1 (2008).
4. F. Jie, K. A. Parton, Mustafid. International Journal
of Logistics Research and Applications 19, 4 (2016).
5. D. A. Efrilianda, Mustafid, I. Rizal..IEEE Xplore
(2018).
(2018).
(2018).
Mathematical Modelling 40, 1 (2016),
9. A. D. Sabila, Mustafid, S. Suryono. In E3S Web of
Conferences 31 (2018).
Production Research (2008)
of Operational Research 256, 2 (2016).
12. R.F. Roldán, R. Basagoiti, L.C. Coelho. J. Appl.
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01 130 32 211
1Master Program of Information Systems, School of Postgraduate Studies, Diponegoro University, Semarang - Indonesia 2,3,4,5Department of Statistics, Faculty of Sains and Mathematic, Diponegoro University, Semarang – Indonesia
Abstract. Product inventory control can be used to provide information related to processes of production
and distribution products based on lead time and product demand from consumers. The research aims to
design product inventory control systems for stochastic lead time demand, and using exponential
smoothing to make lead time demand prediction. The design of the control system uses the probability
method based on the estimated number of product demand from the market or consumer. This research
provides theoretical framework for control systems of products that can determine the size of inventory
parameters in supply chain management, that is reorder point and fixed order quantity, so it can manage the
production amount and supply of products according to market or consumer demand. The benefits of a
product control systems in the inventory management are to provide the necessary inventory allocation
information in making decisions to determine the amount of production and distribution of products to
consumers.
1 Introduction
In the era of quality and business competition is very
tight, the life cycle of clothing products to the consumer
demands with a shorter time and has an element of
uncertainty. This can lead to a high risk in the clothing
supply chain management industry [2].
Currently, the clothing industry market changes
increasingly dynamic and complex, and must face
market changes every day. This should be a management
action to be able to predict and prepare the business for
each day that changes [3]. Dynamic changes in the
clothing industry must be followed by inventory
management that can anticipate business processes that
require various stages, and each stage is accompanied by
an element of uncertainty caused by uncertain market
demand.
In order for the clothing industry to succeed in a
highly competitive business environment, clothing
industry must be able to run supply chain inventory
management in order to improve supply chain
performance continuously [4]. The inventory control
model (ICM) is part of the supply chain inventory
management that plays an important role in controlling
the company resources in all links by paying attention to
the quality of each supplier link to the customer [5, 6].
Many companies are faced with a rigorous, competitive
environment and emerging uncertainties as a result of
information technology innovation and changing
customer needs. Inventory control (IC) can be used for
consistent and sustainable performance measurement of
each link in the supply chain that makes a key role in the
success of a company to achieve a business target goal
with a certain profitability [7].
The ICM must be adaptive to adapt to changes in
market or consumer demand. As an example of a higher
level of consumer demand from inventory will have an
impact on the aspect of customer satisfaction to get the
product according to the desired time. In this case, ICM
aims to overcome the existence of deficiency or excess
product in accordance with production capacity and
inventory [8].
control systems (ICS) in the clothing industry focus on
two variables, that is variables of demand and lead time.
Since both variables are stochastic, modeling the IC
using the probability distribution assumption of both
variables.
normal distribution can yield more suitable results, but
the assumption of a normal distribution of demand data
is often not appropriate, especially if demand per unit of
time or lead time are fluctuating or has a large variance.
The compound distribution of demand during the lead
time is use by the distribution of demand and the
distribution of lead times. The probability distribution of
the combined variables can be normal or other
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
distributions according to the data characteristics of daily
demand and lead time periodically.
The research aims to apply ICS on the predicted lead
time demand (LTD) using probabilistic and statistics
approach. Prediction method in the research using
exponential smoothing approach. The ICS can be used to
determine the numbers of safety stock (SS) and reorder
point (ROP) for planning of product inventory renewal
according to consumer demand.
study of IC with deterministic demand data has been
done by [6] and [9]. The supply chain system framework
is based on information flow in the form of product
demand, and material flow of products has been
developed by [7]. The concept of supply chain system
can be adjusted to be the basis in modeling inventory
control product.
The research designs ICS that can be used to analyze
SS and ROP based on the combined of demand and lead
time stochasticlly. The material studied in this research
includes the use of probabilistic and statistical
approaches in ICM. In this case study, the data used
were not normally distributed, so the exponential
smoothing approach use for transformation into normal
distributed data, and also as a prediction model in the
ICM based on previous actual data of demand.
The variables in this research use the number of daily
demand and lead times, taking a case study in the
clothing industry. Data of daily demand in the form of
historical data from daily demand. The observational
data a sample were taken for 105 days with 8 lead times
from April to June 2016. Daily demand data and lead
times are define as stochastic variables.
The variability of demand and lead time are the basic
for making theoretical development of theory and its
application in IC. SS determination uses a combined of
daily demand and lead time into variables of LTD. In
statistical theory, combined variables are called
compound variables. The LTD is a random variable, in
which those distribution is determined by the combined
of the distribution of demand variable and lead variable.
Data of demand from consumers or transactions with
consumers in retail are monitored through demand data
on supply chains sent by retail every day. The demand
data is input to the system, then processed to produce the
output of estimation of SS and ROP as the basis for
performing the amount of stock order that is useful for
arranging product quantities and product distribution
according to the number of demand.
3 Stochastic lead time demand
The ICM is designed using variable of LTD based on
demand on the lead time from consumers or retail. Let D
be a srandom variable in the form of a daily demand in a
periodic lead time L. Daily demand (D) and lead time (L)
are stochastic random variables. Daily demand and lead
time are assumed to be random variables that are
mutually independent and distributed identically. The
random variable of demand D is also assumed to be
independent with the lead time L. LTD X at time L is
expressed by:
X = (1)
Suppose mean and variance of D and L are respectively
expressed by:
The random variables of D and L are can be
estimated using observation data. The mean values of (2)
and (3) can be estimated by data observations:
= (5)
= (6)
Similarly, the variance values of (2) and (3) can be
estimated by:
= (7)
= (8)
whrere D1, D 2 , …, D N are data of demand with N = L1
+ L2 + · · · + Lk, Li is lead time in i period, i = 1, …, k.
Using the unbiased properties of variable of demand
D and lead time L, then by (5), (6), (7) and (8) obtained:
E( = dan E( ) =
E( = dan E( ) =
the mean and variance values of observational data
(samples) with formulas (5), (6), (7) and (8).
The variable X in equation (1) can be define as a
compound random variable, and mean and variance of X
can be derived by:
= (10)
In certain cases, if D and L have a normal distribution,
then distribution of X can be determined as a normal
distribution. For example, Di is random variable having a
normal distribution with mean and variance , , it
can be proved that X as a new random variable following
the normal distribution with mean and variance
.
The prediction of the number of demand in a certain
period is based on the forecast data from the previous
demand. The prediction model for next demand use
exponential smoothing approach of the demand during
period T [10] with illustration as in Fig. 1 and Fig 2.
Let is the daily demand as stochastic process at
time t. Predicted daily demand D for the next one (Fig 1)
is calculated based on the previous time period T using
exponential smoothing with the formula:
/ T (11)
where is the demand prediction for time t + 1, is
the actual demand at time t, and α is the parameter which
is the weight of the previous demand. The weight value
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
2
of α is derived for data is normal distribution with a low
error.
Fig. 2. Demand prediction for next lead time
Furthermore, using (11) can be calculate the
prediction of demand for period t to lead time L, (or tj +
Lj) by formula:
Based on (12), we can define prediction of all daily
demand with k lead time by:
= (13)
The value of mean and variance from calculated in the
same way by the formula (9) and (10).
The computational results for parameters of the
demand and lead time using the formula (5), (6), (7), (8)
and (9) are given in Table 1. The profile of actual daily
demand data and the smoothing data predicted with
exponential smoothing are given in Fig, 3. Based on the
profile in Fig 3, show that variable of demand and lead
time are probabilistic. Using the formula (12) and (13),
we can calculate the value of total and average of LTD
prediction based on all daily demand and all lead time as
a basis for determining ROP and ROP,
Table 1. Parameter of lead time and LTD
Parameter Lead
satisfies the assumption of mutually independent and
distributed identical. The probability approach is used to
find the probability distribution of LTD. To prove that
the data of demand and lead time have a normal
distribution is used the Kolmogorov-Smirnov test. The
result of normal distribution hypothesis test for actual
demand data gives p-value = 0,026, which states that
actual data of demand is not normally distributed for α =
0,05. Therefore, it is necessary to refine the data with
exponential smoothing method using formula (11).
The smoothing of demand data with exponential
smoothing (11) is given in Fig.3, and based on
Kolmogorov-Smirnov test for the smoothing of demand
gives p-value = 0,200. Using α = 0.05, and since p-value
= 0.200> α = 0.05, thus accepting the hypothesis that the
data of smoothing demand/ predicted of demand is
normally distributed. Likewise for the observation data
lead time, obtained p-value = 0.200, so that lead time is
also normally distributed.
The prediction model with exponential smoothing (11)
can be used for the smoothing model as a transformation
model for normal distribution and as a prediction model.
The prediction accuracy can be determined by the
MAPE (Mean Absolute Percentage Error) method, with
the formula:
MAPE =
where N is the number of daily demand data used for the
prediction model. The computational results of the
predicted data in chart form are given in Fig. 3.
Furthermore, the computational results of MAPE show
that for the prediction of demand obtained MAPE = 45%
(reasonable), with for predicted result for LTD obtained
MAPE = 6% (high accuracy). Although MAPE for data
demand data prediction is quite large, but for data
prediction of LTD is very small.
5 Inventory control systems
information system (SCIS) framework with daily
demand and lead times as the input of systems. Daily
demand and lead time is used to make predictions of
product needs as materials to plan the number of
products to be produced. The prediction model use
exponential smoothing method with equation (11).
The design of ICS for the clothing industry are given
in Fig. 4, with system inputs in the form of daily demand
and lead times (Fig 3). Procedures in access to data input
for the system, the input system in the form of demand
from consumers, initially accommodated in retail, so
accessed into the database system conducted by
administrators and retail. The system database is
designed as an integrated database within ICS.
Fig. 4 shows that the consumer or retail demand data
is the basis for making predictions of the number of
products produced. Predicted product quantities and
product inventory estimates are determined on the basis
of SS and ROP as parameters in IC. Web administrators
have a master data entry facility that is used to estimate
the management of product stock for raw material
planning required in producing a number of planned
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
as SMS / WA every day.
Fig. 4. Inventory control systems
The ICS is defined as a tool for the management and
control of supply, storage, distribution, and recording of
products to maintain adequate stock quantities in
accordance with consumer demand or needs. With this
control system is expected to generate maximum profits
with minimal inventory investment without reducing the
level of customer satisfaction or the level of demand or
order filling. Three parameters used for control
inventory:
Probability approach with normal distribution is used to
make ICM with input data of demand and lead time in
the form of random and stochastic variable. To state that
the stochastic process or random sample data satisfy the
assumption of normal distribution will be use by
Kolmogorov-Smirnov test. With normal distribution
function it can be used to determine SS and ROP as
given in Fig. 5.
Using the properties of the normal probability for
random variable X, then expecting X ≤ ROP, it can be
determined the probability that LTD is less than ROP
with the least probability of 1 - α:
P(X ≤ ROP) ≥ 1 - α
as given in Fig. 5. If using α = 0.05 or 5%, then
probability for LTD is less than ROP given in the
formula:
Therefore, the level of confidence that inventory for
demand is always over 95%. Using the properties of the
confidence interval on the normal distribution curve
(Fig. 5), it is obtained that
ROP = E(X) +
and using formula (16) with α = 0,05, = 1,65,
obtained:
SS =
= 431.
ROP = +
= 1632.
The result of computing for parameter of the ICS is
visualized with the output of a normal distribution curve
showing the mean, SS and ROP as given in Fig. 6.
Fig. 6. Output for SS and ROP
In IC on the clothing supply chain industry, stock
supply monitoring on the supply chain is based on
checking on the current stock status. Current stock is
calculated by:
Current stock = initial stock - total demand
Based on demand data, then at the end of the lead time
can be done the latest stock calculation (end), that is the
initial stock at the beginning of lead time minus the
number of LTD. Furthermore, the latest stock results
compared to SS.
a. No inventory is procured, if the current inventory
number is greater than the number of SS of the
product : IP – LTD > SS.
b. If the current inventory number is smaller than the
number of SS inventory, a procurement demand will
be made, with quantity demand Q is determined by
formula:
where IP is the current stock position, LTD is the
number of LTD, SS is the SS of the product, whereas Q
is the number of product demand. Using the formula
(17), the number of stocks delivery is as follows:
Q = SS -IP + LTD
Stock monitoring is done by looking at the latest
stock system output at lead times compared to SS. If the
current stock is larger than the SS, then the stock is in a
safety condition, meaning no stock renewal is made. On
the contrary, if current stock is smaller than SS, then
stock is in watch condition, that means stock updating
must be done soon, with minimum stock is ROP. In this
case research, for the last lead time it appears that the
current stock is 266 pcs. By comparing the SS = 431, the
current stock is smaller than SS. So as soon as the stock
renewal is equal to Q = 1483 for the next lead time.
https://doi.org/10.1051/e3sconf/201873 , (2018)E3S Web of Conferences 73 ICENIS 2018
01 130 32 211
demand data as a condition of using SS formula (16),
while stocks are safety determinant factor for calculating
stock renewal. The use of the probability method for lead
time demend modeling with a more accurate estimation
of inventory parameters should take into account all
available demand data within a given time period of a
stochastically. In the traditional approach, demand
parameters are estimated taking into consideration only
with demand during the lead period. The length of lead
time can be either constant or deterministic, but the
beginning of the refill process is not known before, so
simply using the demand during the lead time alone can
cause a very bad estimate.
Because demand is stochastic, the number of demand
often changes. This has an impact on SS, ie that SS tend
to be low when demand intensity is low, and sometimes
high if demand intensity is high. To overcome this, the
determination of SS is expected to overcome the
deviation (demand) on a particular day that often occurs.
Many IC research by establishing a safe stock use the
assumption that demand and lead times are deterministic
[10, 11]. Similarly, the number of demand for any given
time period is not known in advance, either the number
of demand or the time of the demand. The problem has
an impact on the results of the analysis that is not
necessarily realistic. Other obstacles in IC, product
delivery time can be uneven and uncertain, for several
reasons, such as the amount of demand with erratic time,
quality and transportation problems.
theory on ICS designed within the framework of SCIS
with probability approach. This result is consistent with
the results of previous research. For example [12] which
states that studying the behavior of demand and lead
time is very important to achieve a useful system
representation in order to make the right decision.
6 Conclusion
framework using input demand and lead time. As a case
study, Inventory control systems can be used to
determine the number of safety stock and reorder point
of lead time demand according to market or consumer
demand. Determination of safety stock and reorder point
using normal distribution curve.
amount to perform a stock renewal.
References
of Operations Management 33, 1 (2015).
2. A. Agarwal, R. Shankar, M.K. Tiwari. Industrial
Marketing Management 36 (2007).
Computer Security 16, 1 (2008).
4. F. Jie, K. A. Parton, Mustafid. International Journal
of Logistics Research and Applications 19, 4 (2016).
5. D. A. Efrilianda, Mustafid, I. Rizal..IEEE Xplore
(2018).
(2018).
(2018).
Mathematical Modelling 40, 1 (2016),
9. A. D. Sabila, Mustafid, S. Suryono. In E3S Web of
Conferences 31 (2018).
Production Research (2008)
of Operational Research 256, 2 (2016).
12. R.F. Roldán, R. Basagoiti, L.C. Coelho. J. Appl.
Log., in press (2016).
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